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A threshold AR(1) model. (English) Zbl 0541.62073

Let \(\{a_ t\}\) be i.i.d. random variables with mean 0. Consider the model \(Z_ t=\Phi_ 1Z^+_{t-1}+\Phi_ 2Z^-_{t-1}+a_ t\), where \(\Phi_ 1\) and \(\Phi_ 2\) are real parameters and \(x^+=\max(x,0)\), \(x^-=\min(x,0)\). It is proved that the process \(\{Z_ t,t\geq 0\}\) is ergodic iff \(\Phi_ 1<1\), \(\Phi_ 2<1\) and \(\Phi_ 1\Phi_ 2<1\). (This is very surprising, since one would expect the condition \(| \Phi_ 1|<1\), \(| \Phi_ 2|<1.)\)
Further, it is shown that the least squares estimators of \(\Phi_ 1\) and \(\Phi_ 2\) are consistent and asymptotically normal, and a test of the hypothesis \(\Phi_ 1=\Phi_ 2\) is proposed. The asymptotic formulas are complemented by small-sample results from simulated data.
Reviewer: J.Anděl

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60J05 Discrete-time Markov processes on general state spaces
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